ORIGINAL RESEARCHpublished: 18 April 2017

doi: 10.3389/fgene.2017.00048

Frontiers in Genetics | www.frontiersin.org 1 April 2017 | Volume 8 | Article 48

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University of Ottawa, Canada

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Universidad Nacional Autónoma de

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Washington University in St. Louis,

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*Correspondence:

Mark A. Ragan

m.ragan@uq.edu.au

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Published: 18 April 2017

Citation:

Taherian Fard A and Ragan MA (2017)

Modeling the Attractor Landscape of

Disease Progression: a

Network-Based Approach.

Front. Genet. 8:48.

doi: 10.3389/fgene.2017.00048

Modeling the Attractor Landscape ofDisease Progression: aNetwork-Based ApproachAtefeh Taherian Fard and Mark A. Ragan*

Institute for Molecular Bioscience, University of Queensland, St. Lucia, QLD, Australia

Genome-wide regulatory networks enable cells to function, develop, and survive.

Perturbation of these networks can lead to appearance of a disease phenotype. Inspired

by Conrad Waddington’s epigenetic landscape of cell development, we use a Hopfield

network formalism to construct an attractor landscape model of disease progression

based on protein- or gene-correlation networks of Parkinson’s disease, glioma, and

colorectal cancer. Attractors in this landscape correspond to normal and disease states

of the cell. We introduce approaches to estimate the size and robustness of these

attractors, and take a network-based approach to study their biological features such

as the key genes and their functions associated with the attractors. Our results show

that the attractor of cancer cells is wider than the attractor of normal cells, suggesting a

heterogeneous nature of cancer. Perturbation analysis shows that robustness depends

on characteristics of the input data (number of samples per time-point, and the fraction

which converge to an attractor). We identify unique gene interactions at each stage,

which reflect the temporal rewiring of the gene regulatory network (GRN) with disease

progression. Our model of the attractor landscape, constructed from large-scale gene

expression profiles of individual patients, captures snapshots of disease progression and

identifies gene interactions specific to different stages, opening the way for development

of stage-specific therapeutic strategies.

Keywords: attractor, landscape, disease progression, gene-regulatory networks, Hopfield networks

INTRODUCTION

Gene regulatory networks (GRNs) regulate diverse biological processes including cell-lineagecommitment and differentiation. GRNs are robust in maintaining their functionality against a widerange of perturbations. Inappropriate regulatory signals can trigger cascades of failures that causeGRNs to malfunction and a disease phenotype to appear (Huang et al., 2009; del Sol et al., 2010).In most instances it is not aberrant activity of a single gene, but rather the perturbation of genenetworks, that drives disease progression (Stower, 2012).

Computational modeling provides insight into GRN dynamics and how the interplay of genescan lead to alternative phenotypes. Onemodeling framework is the attractor landscape. Huang et al.(2005) described a high-dimensional space in which each coordinate represents the expression ofa gene in the GRN. Attractors in this landscape correspond to stable equilibrium states associatedwith a specific cell type (Huang et al., 2005, 2009). Disease phenotypes such as cancer can be viewedas abnormal cell types and represented as latent attractors in this landscape (Huang et al., 2009).

Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

Trajectories across the landscape correspond to developmentalprocesses or disease progression, and elevation (i.e., the z-axis) isinversely proportional to the likelihood of a particular state (GRNconfiguration; Huang et al., 2009; Wang et al., 2010).

Experimental studies have confirmed that attractor landscapesand state-space trajectory models can provide insight intothe biological basis of developmental processes (Huang et al.,2005; Chang et al., 2008; Huang, 2009). Other frameworkshave been employed to model such state-space trajectories andattractor landscapes, including model-free approaches (Chenet al., 2012); logical models such as Boolean networks and Petrinets, which provide a quantitative model of a GRN, allowingusers to gain an overall understanding of the behavior ofthe system under different conditions; and continuous modelssuch as those based on ordinary differential equations (ODEs)or continuous linear models, which provide a framework tocapture and understand stochasticity in the system using real-valued molecular concentrations (rather than discretized values)over a continuous time scale (Karlebach and Shamir, 2008).In the context of cancer, Saez-Rodriguez et al. (2011) usedlogical models to compare normal and transformed hepatocytenetworks; Esfahani et al. (2011) proposed an algorithm basedon Boolean networks with perturbation, and through partialknowledge of the GRN and gene-expression values reconstructedtumor progression; and Lucia and Maino (2002) used ODEs tomodel the interaction of tumors with the host immune system.These frameworks tend to be based on small gene regulatorycircuits, and/or require extensive prior knowledge of the system(Maetschke and Ragan, 2014; Taherian Fard et al., 2016).

If disease is viewed as a pre-existing configuration of theGRN, accessed via specific mutations or other changes to thesystem (Huang et al., 2009), one can model trajectories ofdisease progression by using an appropriate time-course gene-expression profile. However, given the scarcity of homogenousor isogenic samples (as samples come from different patients),lack of dynamic experimental data, and very limited time-coursedisease progression gene-expression data, to date it has not beenfeasible to construct a comprehensive model of GRN dynamicsin disease.

Here we employ the mathematical formalism of Hopfieldnetworks (HNs; Hopfield, 1982) to construct and visualize thelandscape of disease progression, based on large-scale gene-expression profiles from patients with different stages of diseaseor cancer grades. We characterize normal and disease states ofthe cell as attractors of Hopfield networks; estimate their sizeand robustness; and take a network-based approach to identifythe unique biomolecular interactions that underlie each stage ofdisease progression. These attractors correspond to local minimaof an energy function, and are formed by iteratively updatingthe network; transient states correspond to intermediate time-points in the gene-expression profile, while trajectories tracethe convergence of samples to their attractors. We hypothesizethat an attractor with a large basin is more likely to attract amore-heterogeneous set of samples. To remain as close to thebiology as possible, we utilize a correlation network (computedfrom large-scale time-course gene-expression data) to constructthe model. Changes in this correlation network through time

correspond to rewiring of the GRN through the stages of diseaseprogression.

MATERIALS AND METHODS

Hopfield NetworksWe used Hopfield networks (HNs; Hopfield, 1982) to modeltrajectories of disease progression and track rewiring of theunderlying GRN. AnHN is a fully connected neural network withnodes i ∈ 1,..., n and undirected edges wij between nodes i and j,representing genes and their interaction, respectively. There aretwo major steps involved in constructing the HN. Firstly in thetraining phase, we construct the weight matrix W based on thePearson correlation coefficient (PCC) between the gene pairs.Wis a symmetric matrix, withwij = PCC (i,j) andwii = 0 for nodes iand j. Secondly, in the recall phase, dynamics of the network andconvergence to the attractors are defined by the product of thepattern matrix (gene expression profile) and the weight matrixW (bottom panel in Figure 1).

P(t+1) = sgn(

P(t)W)

(1)

where P(t) is the state of the pattern (here samples) at time step t,followed by a discretization through a sign function.

The energy E is computed through a family of monotonicallydecreasing functions, in this case the Lyapunov function thatguarantees convergence to a low-energy attractor state.

E[P(s)] = −1

2PWPT (2)

where E[P(s)] is the energy of the network state s at time t, forpattern P.

We do not provide sample labels to the algorithm; thepattern learning step is non-parametric, that is we leave it tothe algorithm to construct attractors based on similarities amongthe patterns. To visualize the landscape in three dimensions,we interpolated the energy values over a two-dimensional gridconstructed from the first and the second principal componentsof the dimensionally reduced gene-expression data. For moredetails please refer to Taherian Fard et al. (2016). This workdiffers from and extends our previous methods as follows: (1)here the term “attractor” refers to a Hopfield attractor that isgenerated as the result of an iteration process and correspondsto a local minimum of the energy function; (2) the perturbationanalysis is carried out directly on the W by assigning randomvalues to the edges in the network; and (3) we introduce methodsto measure the width and depth of attractors. All HN analysiswere performed on a standard workstation (Windows OS) andcompleted in <5 s for each dataset.

Estimating the Size and Robustness of anAttractorWe estimated the size of attractors by measuring their widthand depth. To estimate the width of an attractor, we computedthe intra-group distance of all the samples converging to theattractor. There are different approaches to calculate the intra-group distance of the elements in a group, including sum,

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

FIGURE 1 | The workflow. The top panel describes the correlation network analysis through which we identified unique genes and interactions at each disease

stage, and biological characteristics of stage-specific correlation networks. The bottom panel shows the steps taken in constructing the network through which we

visualized the landscape and estimated the size and robustness of the attractors.

minimum, maximum or average pairwise distance between allpoints in a group, or between the centroid and all pointsin the cluster. To obtain a comparable measure of widthacross all attractors we used the average standardized pair-wiseEuclidean distance of samples converging to the same attractor.This estimate provides a quantitative measure of the variationof samples converging to a specific attractor. The depth wasmeasured by calculating the energy difference of samples beforeand after convergence.

Although the elements of the GRN remain the same, theirinteractions change from normal to disease phenotype. In ourmodel, the W matrix holds information on the interactionsbetween the entities of the network. Therefore, in order toassess the effect of perturbations on the network, we randomlyperturbed 50% of the edges in W. From our previous study(Taherian Fard et al., 2016) we know that at 50% perturbation, thenetwork does not reflect a stable phenotype, but enough signalremains (i.e., has not been randomized away) consistent with thenetwork representing a transient or unstable cell state in a livingsystem. The W matrix perturbation step was then followed bythe HN recall phase using the perturbed W, computation of the

fraction of the samples that did not converge to their respectiveattractor, and computation of the Hamming distance (HD) of thesamples to their attractor after each iteration. The HD betweenthe two binary strings indicates the proportion of values thatdisagree between them. The highest difference is observed at HD= 1. At HD= 0, the strings are identical. The Hamming distancewas computed using SciPy tool 0.15.0 (http://www.scipy.org) inPython.

DatasetsThe first case-study (DeMarshall et al., 2015) identifies candidateblood-based autoantibody biomarkers that are useful for earlydetection and diagnosis of Parkinson’s disease (PD). ProtoArrayv5.0 Human ProteinMicroarrays (Invitrogen, Carlsbad CA) wereused to identify differentially expressed autoantibodies in humanserum samples. With an overall accuracy of 97.5%, the candidatebiomarkers were capable of distinguishing early-stage PD fromadvanced PD. The dataset (GEO accession number: GSE62283)encompasses 45 human serum samples, 15 for each disease stage:control (normal), early PD (EPD), and advanced PD patients(Table S1).

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

The second case-study (Sun et al., 2006) investigates the effectof stem cell factor (SCF) expression in human gliomas in a grade-dependant manner. SCF induces angiogenic response in vivo bydirectly activating brainmicrovascular endothelial cells. Sun et al.(2006) found that SCF overexpression is associated with poorprognosis in glioma patients, whereas its down-regulation resultsin improved survival in mouse models. SCF is up-regulated inhigh-grade gliomas and down-regulated in non-tumor samples,making it a potential anti-angiogenic target inmalignant gliomas.The dataset (GEO accession number: GSE4290) includes mRNAexpression data (Affymetrix Human Genome U133 Plus 2.0Array) from patients with different grades of glioma. The 96samples include 23 non-tumor samples (from epilepsy patients),38 grade II, 12 grade III, and 23 grade IV gliomas (Table S1).

The third case-study compares the gene-expression profiles ofprimary tumors with and without distant metastasis, to identifycandidate genes that influence the prognosis of patients withcolorectal cancer (CRC). Using real-time reverse transcriptionPCR, Matsuyama et al. (2010) found that grade II and grade IIICRC patients with low expression of MUC12 showed the worstdisease-free survival, suggesting prognostic value for MUC12in postoperative adjuvant therapy for these patients. There are17 normal (from adjacent tissues), 47 non-metastatic, and 30metastatic samples in this dataset (GEO accession number:GSE18105; Table S1).

Each dataset was z-score normalized (µ = 0, σ = ±1),followed by feature selection to extract probes with the highestvariation across groups.We ranked genes based on their variance.The index beyond which the variance is essentially unchanged(that is, the elbow of the variance-over-feature plot) was used asa cut-off to select the number of features (genes).

Biological Properties of Stage-SpecificNetworksCorrelation network were constructed for each disease stage, andcommon genes and interactions across groups were identified.We used the first 100 feature-selected genes to construct thecorrelation networks, and chose the genes with significant (p< 0.0001) interactions for further analysis. The p-values weregenerated using standard t statistics and were generated as asignificance measure while computing the PPC. We subtractedthe set of common genes and interactions from the originalnetworks to obtain a network of genes with unique interactionsspecific to each disease stage (top panel in Figure 1).

We used Ingenuity Pathway Analysis (QIAGEN, RedwoodCity CA) for gene function and biological network analyses.Statistical analysis was performed using R 3.0.1. Correlationnetwork comparisons and visualization were performed usingCytoscape 3.2.1 (Shannon et al., 2003).

RESULTS

Our proposed model provides a framework to analyse the overallbehavior of a GRN during progression from a normal to a diseasestate. Figure 2 shows a 3D view of the Hopfield energy landscapeconstructed from each of our three datasets, while Figure 3 shows

the effect of perturbations on the shape of these landscapes. Foreach disease-specific correlation network, and for the respectivenormal and disease attractor networks, we identify the GeneOntology (GO) biological process (BP) terms and functions, andthe genes uniquely associated with each network (Figure 1). Theidentities of proteins with unique interactions in each grade-specific network are presented in Table S2.

Case Study 1: Parkinson’s Disease (PD)ProgressionThe first dataset (GSE62283) consists of 45 samples and 9,480human protein-probes (Table S1), of which 118 probes remainafter feature selection (refer to Datasets in section Materialsand Methods). Normal samples exhibit the highest average E-value (Enormal = −120); the energy values are similar betweennormal and Parkinson’s disease samples (EEPD = −122), but theenergy decreases substantially at mid-stage PD (EPD = −278;Table 1). The energy difference between normal samples beforeconvergence and the normal attractor (1E = 1,329) is greaterthan that between advanced-stage PD samples and the diseaseattractor (1E = 1,173), i.e., the normal attractor is deeper(Table 1); and the average pairwise intra-group distance is greaterfor the normal attractor (1.72) than for the one representing PDcases (1.52), i.e., the normal attractor is wider as well.

We tested the robustness of each attractor by randomlyperturbing a large proportion (here 50%) of the edges inW, thenallowing each sample to relax to its attractor. We then countedthe number of samples that did not converge. For the PD data,the normal attractor was more resistant to perturbation: 20% ofsamples failed to converge, whereas 30% of the correspondingsamples did not converge to the disease attractor (Figure 4 andFigure S1).

In this case study, we used all feature-selected protein-probesto construct the stage-specific networks. The correlation networkof the PD group had the greatest number of interactions (598edges), followed by the EPD (186) and normal networks (123).These networks do not have any interactions in common. Nodenumber varied in the unique networks: the PD group had thegreatest number of nodes, followed by the normal and EPDgroups. Table 2 shows the numbers of positively and negativelyco-regulated interactions, and the number of unique interactionsneeded for the network to progress from one state to the next.

The genes encoding proteins with unique interactions in thenormal correlation network were slightly enriched for the GOterm receptor-mediated endocytosis (p= 1.3E-2; Table 3). Amongthe unique genes associated with this network is nucleosidediphosphate kinase 7 (NME7). The protein encoded by this genehas a major role in synthesis of nucleoside triphosphates otherthan ATP. Garcia-Esparcia et al. (2015), investigating the effectof NME7 and other genes involved in purine metabolism in PD,interpreted the down-regulation of these genes (mainly expressedin neurons) in the midbrain as a consequence of dopaminergiccell death in PD (Garcia-Esparcia et al., 2015).

Positive regulation of calcium-mediated signaling (p = 5.2E-2) was associated with genes in the EPD correlation network(Table 3). Chemokine C-C motif ligand 17 (CCL17) is one of

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

FIGURE 2 | Hopfield energy landscape for the three case studies in 3D (left column) and 2D (right column): top, Parkinson’s disease; middle, gliomas;

bottom panel, colon cancer. The x- and y-axes are respectively the first and the second principal components of the autoantibody expression data, while the z-axis

is the Hopfield energy value. Each dot represent a sample, colored by disease stage; and the large green dots represent the attractors. The lines in the 2D view

represent the trajectories of samples converging to their respective attractors.

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

FIGURE 3 | Effect of perturbations on the shape of the landscape: top, Parkinson’s disease; middle, gliomas; bottom panel, colon cancer. The red-blue

surface shows the original network, while the gray-scale surface shows the landscape after perturbation. The brackets show the difference in attractor E-value before

and after perturbation.

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

TABLE 1 | Mean energy of samples in their original state i.e., before

convergence.

Data Group E E dis. attract†

Parkinson disease Normal −120 1329

Early PD −122 1329

PD −278 1173

Glioma Normal −536,922 332,579

Grade II −240,789 628,712

Grade III −306,800 562,701

Grade IV −398,300 471,201

Colon cancer Normal −786,802 258,735

Non-metastatic −197,287 848,250

Metastatic −160,845 884,692

†Energy-value distance from attractor.

the unique genes present in this network. Chemokines and theirreceptors are expressed in cells of the central nervous system andare involved in synaptic transmission. Studies have confirmedtheir role in neurological diseases such as Alzheimer’s and PD,making them potential therapeutic targets (Mines et al., 2007).

The GO term Histone H4-K12 acetylation (p = 2.6E-2) wasassociated with genes in advanced PD (Table 3 and Table S4).Among the unique genes present in the network is the FYVE-finger-containing phosphoinositide kinase PIKfyve. The proteinencoded by this gene regulates endosome and lysosome functionby internalizing and degrading voltage-gated Ca2+ channels(VGCCs). Knockdown of PIKfyve prevents the degradation ofVGCCs and thus Ca2+ overload and excitotoxicity in neurons,which have been found to play an important role in PD (Tsurutaet al., 2009).

Genes in the normal attractor network were associated withamino acid metabolism (p = 2.7E-4) and Nur77 signaling inT lymphocytes (p = 7.1E-5) canonical pathways. The genes inthe disease attractor network were enriched for carbohydratemetabolism (p= 3.1E-3) and associated with the calcium-inducedT lymphocyte apoptosis pathway (p = 1.2E-4; Table S3). Aprogressive loss of dopaminergic neurons is the main cause ofParkinson’s disease; and different types of programmed cell deathand signaling pathways and mitochondrial fragmentation havebeen associated with PD (Venderova and Park, 2012).

Case Study 2: Grade-Based Progression ofGliomasThe second dataset (GSE4290) includes 96 samples and 54,613probes, of which 2,859 remain after feature selection (Table S1).Normal samples in their original state (i.e., before convergence)have the lowest energy value (E-value) Enormal = −536,922,indicating tighter correlation between the genes in the underlyingnetwork. We observe an increase in E-value for grades II and IIIgliomas (EG2 =−240,789, EG3 =−306,800) and a lower E-valuefor the final stage of the disease EG4 = −398,300 (Table 1). Theenergy difference between samples before and after convergencewas greater for the cancer attractor (1E = 471,201) than for

the normal attractor (1E = 332,579), implying that the cancerattractor is deeper than the normal one (Table 1). The averageintra-group distance is greater for the cancer (1.79) than for thenormal attractor (1.56), suggesting that the former can attract amore-heterogeneous set of samples.

Upon perturbation the cancer attractor is robust, as samplescontinue to converge even after perturbation of 50% of theedges in the underlying network. This contrasts with the normalattractor, for which the same level of perturbation results in 20%of the samples failing to converge to the corresponding attractor(Figure 4).

Analysis of the underlying correlation networks shows that thenormal-stage network has the highest number of significant (p≤0.0001) interactions (1,361 edges), followed by the grade II, IV,and III networks. Next, we filtered out common interactions (88common edges) from the original networks, yielding networksof genes with unique interactions. The normal network hadthe highest number of unique interactions (1,273 unique edges)followed by grades II, IV, and III (Table 2).

For the normal group, the genes with unique interactions areenriched for GO terms including transmission of nerve impulse(p = 1.0E-5; Table 3 and Table S4). One of these is neurexin3 (NRXN3). The protein encoded by NRXN3 functions as areceptor and cell-adhesion molecule in the nervous system.NRXN3 is highly expressed in normal tissues and down-regulated in human gliomas; moreover, Sun et al. (2013) foundthat Forkhead box Q1 (FoxQ1) promotes glioma proliferationby down-regulating NRXN3, suggesting that it may be a tumorsuppressor gene (Sun et al., 2013).

Gamma-aminobutyric acid (GABA) signaling (p= 4.9E-5) wasassociated with genes in grade II gliomas (Table 3 and Table S4).GABA is the main inhibitory neurotransmitter in the centralnervous system, and has been shown to regulate the growth ofmany cell types including neuronal and tumor stem cells. GABAresponse is associated only with low-grade gliomas, suggestingthat its absence results in unlimited growth of malignant gliomas(Smits et al., 2012). MET proto-oncogene tyrosine kinase (MET)was among the genes with unique interactions in this network.Following activation by hepatocyte growth factor (HGF) ligand,it triggers cascades of signaling pathways including RAS-ERKand PI3 kinase-AKT. Mutation in MET is associated with tumorgrowth, angiogenesis and metastasis. Amplification of MET andits ligand HGF has been associated with primary and lower-gradegliomas (Fischer et al., 1997; Beroukhim et al., 2007).

Genes in the grade III correlation network were enrichedfor neurological system process (p = 1.0E-3; Table 3 andTable S4). Neurotensin receptor 2 (NTSR2), which encodesa G-protein-coupled receptor, was among the genes withunique interactions in this network. Neurotensin and itsreceptors including NTSR2 play an important role in oncogenicprogression of cancer malignancies. Activated NTSR2 is the keyregulatory component that promotes the phosphorylation ofextracellular signal-regulated kinase 1/2 (ERK1/2) in glioma cells.ERK1/2 can mediate cell proliferation and apoptosis throughoverexpression of PDGFRA—a type of receptor tyrosine kinasewith ERK-dependant activity (Chen et al., 2014; Ouyang et al.,2015).

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

FIGURE 4 | Distance of samples from their respective attractors at each iteration in the original network, and the perturbed network at 50%

perturbation rate. Left, Parkinson’s disease; middle, gliomas; right panel, colon cancer. For each disease-normal pair, comparing the median-line and boxes shows

how the attractors differ in size.

Synaptic transmission (p = 1.9E-5) is one of the GO termsassociated with grade IV tumors (Table 3 and Table S4).Among the genes with unique interactions in this group isepidermal growth factor receptor (EGFR), which encodes amember of the protein kinase superfamily. Following receptoractivation by binding to a ligand, a series of signalingcascades initiates and drives many cellular responses includingcell proliferation and an anti-apoptosis process. EGFR isassociated with higher-grade gliomas (Kunkle et al., 2013),and its amplification and activating mutation can be accuratemolecular markers in glioma subtyping (Brennan et al.,2009).

We performed similar analyses for the samples that convergedto the normal and the cancer attractors. GABA receptor signaling(p = 2.7E-05) was among the canonical pathways associatedwith the samples converged to the cancer attractor. Wnt/cateninsignaling, associated with the normal group (p = 1.1E-2), playsan important role in glioma proliferation and tumor progression(Nager et al., 2012; Chen et al., 2013). Cell death and survival(p = 1.99E-04) was enriched for the cancer attractor, whereascellular growth and proliferation (p = 1.36E-4) was among themolecular cellular functions associated with the normal attractor(Table S3).

Case Study 3: Colon Cancer Progressionfrom Normal to Metastatic StateThe third dataset (GSE18105) contains a total of 94 samples and54,675 probes (Table S1) of which 3,960 probes remained afterfeature selection. Normal samples at their original state resideat the Enormal = −786,802 energy level. Non-metastatic andmetastatic samples energy values were Enon−metastatic =−197,287and Emetastatic =−160,845, respectively (Table 1). Analysis of thedepth of attractors showed that the metastatic attractor is deeperthan the normal attractor as the energy difference of samples

TABLE 2 | Comparison of group-based correlation networks: first 100

feature-selected genes with p ≤ 0.0001.

Data Group Original network Unique network

Nodes Edges (±) Nodes* Edges (±)

Parkinson

disease

Normal 118 123 (91/32) 65 123 (91/32)

Early PD 118 186 (173/13) 61 186 (173/13)

PD 118 598 (367/231) 96 598 (367/231)

Glioma Normal 100 1361 (1319/42) 68 1273 (1256/17)

Grade II 100 1144 (1090/54) 83 1056 (1027/29)

Grade III 100 101 (75/26) 18 13 (12/1)

Grade IV 100 988 (906/82) 84 900 (843/57)

Colon

cancer

Normal 100 215 (131/84) 47 86 (58/28)

Non-metastatic 100 200 (113/87) 58 71 (40/31)

Metastatic 100 139 (81/58) 18 10 (8/2)

*The genes that remained with no interaction after the filtering process are removed from

the network.

before and after convergence is higher in the cancer attractor(metastatic vs. normal attractor = 884,692 vs. 258,735; Table 1).Estimates of the size of attractor indicated that the cancerattractor has a greater width, as the average pair-wise Euclideandistance of all samples is larger than the normal attractor (cancervs. normal = 1.73 vs. 1.69). Perturbation analysis revealed thatboth attractors are equally robust, as in both cases 10% of thesamples do not converge to their respective attractors (Figure 3).

The correlation network analysis showed that the normalnetwork had the highest number of edges (215), followed by thenon-metastatic and then metastatic correlation networks with200 and 139 edges respectively. After removing the commonedges, the normal correlation network exhibited 47 nodes and 86unique edges, followed by the non-metastatic (58 nodes and 71

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TABLE 3 | Functional analysis of group-based correlation networks (based

on genes with unique interactions in each network)*.

Data Group GO term p-value

Parkinson

disease

Normal Receptor-mediated endocytosis 1.3E-2

Early PD Positive regulation of calcium-mediated

signaling

5.2E-2

PD Histone H4-K12 acetylation 2.6E-2

Glioma Normal Transmission of nerve impulse 1.0E-5

Grade II Gamma-aminobutyric acid signaling 4.9E-5

Grade III Neurological system process 1.0E-3

Grade IV Synaptic transmission 1.9E-5

Colon

cancer

Normal Ion transport 9.6E-2

Non-metastatic Multicellular organismal process 2.7E-2

Metastatic Development process 3.2E-2

*Please refer to Table S4 for the full list of GO terms (p ≤ 0.0001).

edges) and the metastatic correlation network (18 nodes and 10edges) with the fewest genes and unique interactions (Table 2).

The genes with unique interactions in the normal networkwere enriched for ion transport (p = 9.6E-2; Table 3 and TableS4). Prostate cancer susceptibility candidate (PRAC) is amongthe unique genes in this network. It is specifically expressed inthe human prostate, rectum and distal colon and has been shownto have a regulatory role in the nucleus (Liu et al., 2001).

The non-metastatic network was enriched for multicellularorganismal process (p = 2.7E-2; Table 3 and Table S4) includingthe calcium activated chloride channel A1 (CLCA1). It isexpressed mainly in the colon, intestine, and appendix. CLCA1plays a role in tumor suppression and has been shown to havea down-regulated expression in colorectal cancer (Yang et al.,2013).

The genes in the metastatic network were associated withdevelopmental process (p= 3.2E-2; Table 3 and Table S4). Amongthe unique genes in this group is IGF2BP3, a member of mRNAprotein binding family and an oncofetal protein that regulatesexpression of genes involved in tumor cell proliferation, chemo-resistance and metastasis. In vitro studies have revealed thatIGF2BP3 up-regulated expression enhances tumor growth, drug-resistance and metastasis in various human cancers (Ledereret al., 2014).

The genes with unique interactions in the normal attractorcorrelation network were enriched for B cell developmentpathways (p = 3.5E-3) and cell morphology (p = 1.06E-5)for molecular and cellular function. In the case of the cancerattractor correlation network, the genes were enriched for cellularmovement (p = 2.0E-4) for cellular function and associated withthe autoimmune thyroid disease signaling pathway (p = 3.6E-3; Table S3). Thyroid hormone signaling has been shown tobe a major factor in digestive system growth, and homeostasisand the expression of thyroid-hormone receptors has beenassociated with colon cancer progression. Other studies havesuggested that thyroid-hormone signaling may suppress coloncancer invasiveness (Brown et al., 2013).

DISCUSSION

Large-scale multi-omics studies have been carried out toinvestigate the molecular dynamics underlie complex diseaseprogression. For instance, in a recent study Cho et al. (2016)used Boolean networks to construct the attractor landscape ofcolorectal tumorigenesis from previously published canonicalsignaling pathways. Combined with already known mutationdata and prior knowledge of signaling networks in cancer, theirmodel provides a new approach for discovering novel therapeutictargets for cancer patients (Cho et al., 2016). Parsons et al. (2008)provided a genetic landscape of glioblastomas by integratingmutations and copy number alternations by sequencing 20,661protein-coding genes, and performed gene expression analysisin 22 human tumor samples. They inferred key genes associatedwith glioblastoma including IDH1, and showcased the potentialof genome-wide genetic studies in opening novel avenues in braincancer research (Parsons et al., 2008).

Attractor landscape models provide a quantitative approachto understand the dynamics of GRNs during developmentand disease formation. Here we construct disease-progressionlandscape models using the mathematical framework of Hopfieldnetworks. We used three datasets to capture different aspectsof this process: the advancement of Parkinson’s disease fromnormal to an advanced state, the progression of gliomafrom normal to grade IV, and the progression from normalcolon to metastatic colon cancer. For each, we mappedthese different stages of disease progression to HN energyprofiles. Attractors in these landscapes correspond to theinitial (normal) state of the cell, and the (potentially) finalstage of the disease. Each landscape is constructed usingthe correlation network among all feature-selected protein-or gene-pairs in the microarray data, reflecting the biologicalactivity and changes in the GRN during disease progression.We take a network approach to identify unique molecularinteractions at each disease stage, and examine their biologicalfeatures.

Complex disease can often be attributed to inappropriateregulatory signals, together with accumulation of geneticmutations. It is important to identify specific genes whosemutation results in moving cells from normal to a disease state,and eventually to disease progression, characterized in cancer byacquisition of metastatic potential. With a view of disease as apre-existing configuration of the GRN that has not been accessedby a normal cell, this model provides a systems view on GRNrewiring during disease formation and progression.

Unlike other computational models of attractor landscapes,our model does not require prior knowledge of the system.Because the Hopfield energy values are computed via a Lyapunovfunction, they are guaranteed to converge to a local energyminimum that represents a stable state of the network orattractor. In this formalism, the energy of samples at theiroriginal state (i.e., before convergence) is inversely proportionalto the tightness of the correlations in the network: the more-correlated the expression of feature-selected genes, the lowerthe energy (Taherian Fard et al., 2016). We emphasize that wedo not provide labels for samples (non-parametric learning),

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Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

and the learning algorithm constructs attractors solely on thesimilarities among the patterns in the gene-expression data.Samples with similar activity patterns typically converge to thesame attractor, but are occasionally misclassified (i.e., converge toa different attractor). The difference between energy values beforeand after convergence provides an estimate of the depth of theattractor.

As discussed above, computational models of GRNs havetheir own definition of attractors and trajectories. Here wecapture trajectories of disease progression through the energyvalues of samples before convergence (color-coded in the 3Dlandscape, Figure 2). Attractors are the result of an artificialupdating of the network (the iterative HN process) and do notcorrespond to actual cell types, but the computed energy ofsamples at their attractor state provides a quantitative measureof the extent of gene-expression similarity among the samplesat the corresponding stage of disease progression. Thus, inthe PD case study, we observe that normal and EPD samplesconverge to the normal attractor, whereas PD samples convergeto the disease attractor. By contrast, in both cancer case studieswe observed high intra-group heterogeneity within the mid-stage subgroups: a subset of grade II gliomas converge to thenormal attractor while the rest converge to the cancer attractor,while a subset of non-metastatic CRC tumors converges tothe normal attractor while the rest converge to the diseaseattractor.

Our results indicate that in the case of PD, the normal attractorhas a slightly broader basin of attraction than does the diseaseattractor, whereas the opposite is true for each of the cancercase studies. Thus, samples extracted from tumors are moreheterogeneous than the corresponding normal tissues, whereasthere is a relatively narrow basin of attraction in PD. It remainsto be seen whether this result will prove general.

A stable attractor is robust to perturbation, such thatintroducing noise into the network does not affect the fractionof samples converging to that attractor. By contrast, a less-stableattractor does not bear the insult. The biological interpretationis that if the attractor is robust, more molecular changes tothe network are required to move cells out of the phenotypicstate. The robustness of attractors proved to be specific toeach case study, and was influenced by factors includingthe presence of nodes strongly correlated with many othernodes.

For the HN framework to have utility, the input datamust meet certain requirements: at least three time-pointsmust be available, with at least three samples per time-point,and samples must be sufficiently homogeneous. As we use acorrelation measure to constitute the edges, a larger samplesize provides a stronger signal for the gene-activity pattern.To capture the dynamics of GRN at well-defined stages ofdisease progression, we chose publicly available datasets withdifferent characteristics. Thus, in the first case study we focusedon severity of disease by using human protein microarraydata from patients with early- and advanced-stage PD. In thesecond case study, we used gene-expression profiles of patients

with different cancer grades. We utilized the graded gliomasamples, assuming that normal and lower-grade gliomas are

precursors of higher-grade gliomas. Similarly in the third casestudy, we targeted the metastatic process in colon cancer,assuming a progression from normal to non-metastatic diseaseto metastasis.

Our HN landscape model, with a simple computationalworkflow and requiring minimum prior knowledge of theunderlying network, provides a framework to study the processof disease formation and progression, and identifies genesthat are potential key drivers of this process. The feature-selected gene sets are informative on the disease states perse, while still supported by the evidence we present in thispaper that the genes with unique interactions at each stage-specific network of the disease are the potential key drivers ofdisease progression. We show that by applying this frameworkto appropriate disease stage-specific protein or gene microarraydatasets, biologically useful insights can be revealed fromthe dynamics of the GRN, including the identification ofgenes and their stage-specific interactions that are involved indisease progression. In constructing GRN models, it could bedesirable to include data on other molecular processes that arealso involved in driving the cell toward a specific fate, e.g.,histone modifications, copy-number variation, and time-courseproteomics data; or in the context of disease, specific mutationsand their downstream regulatory consequences. The Hopfieldmodel framework is sufficiently flexible to utilize and integratea variety of molecular data, so long as at least two states orconditions are present; input data need not be quantitative,e.g., could represent the presence or absence of a histone mark.Combining all layers of information will allowmore-accurate anddetailed modeling of these dynamic systems of development ordisease.

AUTHOR CONTRIBUTIONS

AT and MR conceptualized the research; AT designed andperformed the analysis; AT and MR wrote the manuscript. Bothauthors approve its submission for publication.

FUNDING

Funding was provided by Australian Research Council awardDP110103384 to MR. This project was supported in part bystrategic funds from The University of Queensland.

ACKNOWLEDGMENTS

We thank Dr Alison Anderson for valuable comments.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be foundonline at: http://journal.frontiersin.org/article/10.3389/fgene.2017.00048/full#supplementary-material

Frontiers in Genetics | www.frontiersin.org 10 April 2017 | Volume 8 | Article 48

Taherian Fard and Ragan Modeling the Attractor Landscape of Disease Progression

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